user14033: Computer chess engines have gotten better since Deep Blue beat Kasparov in 1997.
Did the algorithms get better, or were the improvements mostly due to the same algorithms running faster thanks to faster hardware, etc.?
If the former, are these algorithmic improvements public?
And if so, what were the improvements? Where can I read about them?

user4418: 
Did the algorithms get better?

Obviously, yes a little bit.

or were the improvements mostly due to the same algorithms running faster
  thanks to faster hardware and software?

Minor nit: If the algorithms got better then that is the software getting better so there is no "or".
Moore's Law tells us that processor speed will double roughly every 18 months. That means it has doubled roughly 13 times in 20 years. That makes modern processors somewhere in the region of 8,000 times faster. So, far and away the biggest improvement in engine performance is due to faster hardware.

If the former, are these algorithmic improvements public?
And if so, what were the improvements? Where can I read about them?

Well, it wasn't the former, it was the latter. Nevertheless the improvements are mostly open source and freely visible by downloading the sources for engines like Stockfish. Perhaps also worth giving the general Stockfish download link since the specific source code link will likely expire when version 9 comes out.

user12449: I can't speak for the algorithm used for Deep Blue, but I'm going to try and explain the improvements in chess programming.  Speed is the greatest improvement.  Deep Blue used multi-processor dedicated computers, so a comparison isn't really possible.
https://chessprogramming.wikispaces.com/ is a great source, but it's hard to navigate.  
There are 3 main functions that are tweaked to improve a chess engine are the evaluation, move generation, and search functions.
Evaluation is the hardest to program, as there are many exceptions to the rules.  With hard drive space getting cheaper, the eval function allows for more exceptions to be evaluated.
Move generation, along with making and unmaking a move, consumes a lot of memory because it has to be preformed so many times.  The most common generation functions are mailbox, bitboard, 0x88, 8x8, extended boards (10x10, 10x12), and a predetermined move array/table  (*I use an indexed move table).  Current opinion is that bitboards are the faster, and using magic bitboards speed this by up to 30%.  Dr. Robert Hyatt, professor and creator of cratfy chess engine, claims no significant speed increase.
The early search function was the primitive min-max functions.  Basically were you try to maximize the score of the side to move and minimize the opponent's score.  Alpha-Beta was the first improvement.  They reduced the number of moves being searched by transposition table, cut-off values, aspiration windows, and history heuristics.  These are depth-first searches.  There is also the internal iterative deepening search which tries to search the "best" move(s) the deepest hoping that searching other moves will prove to be fruitless.
NOTE:  My index table.
GNUChess and Jester both use an index array to generate their moves.  They initialize the engine by filling array with possible moves.  The take the six pieces and compute the legal moves that are available from each square.  So each piece had a [64][8] array.  I took this idea and compressed it to two indexes and a table.  The table holds a value which tells if the 16 moves are possible, one index holds the offset of the move, and the other holds the mask.
offset[] = {-8, -1, 1, 8, -9, -7, 7, 9, -17, -15, -10, -6, 6, 10, 15, 17};
mask[]   = {1, 2, 4, 8, 16, 32, 64, 128, 256, ...};
Then the generation of a sliding move is as easy as looking up the validity of it's mask in it's allowable offsets against the move table.

user4588: Maybe you can take a look at TalkChess, a forum dedicated to computer chess. I found a recent thread that might be interesting for you: Progress in 30 years by four intervals of 7-8 years
A couple of matches between (former) top engines are played on the same hardware. The test suggests that in the recent years (2002-2017), the gain is mainly made by software improvements. In the test, Stockfish (2017) scored an impressive 94/100 against RobboLito (2009), while RobboLito, on its turn, crushed Shredder (2002) with 92/100.
An important remark: as parallel computing is not implemented in the older engines, the test was performed on a single core. As a result, the hardware gain by parallel machines is not measured. On the other hand, you could argue that parallel computing is also a software gain: it is not easy to design and implement an efficient and well-scaling parallelization for the search algorithm.
The Stockfish engine is open source, so the algorithmic improvements are public. A lot of documentation can be found on https://chessprogramming.wikispaces.com

14033: <p>Computer chess engines have gotten better since Deep Blue beat Kasparov in 1997.</p>

<p>Did the algorithms get better, or were the improvements mostly due to the same algorithms running faster thanks to faster hardware, etc.?</p>

<p>If the former, are these algorithmic improvements public?</p>

<p>And if so, what were the improvements? Where can I read about them?</p>
 4418: <blockquote>
  <p>Did the algorithms get better?</p>
</blockquote>

<p>Obviously, yes a little bit.</p>

<blockquote>
  <p>or were the improvements mostly due to the same algorithms running faster
  thanks to faster hardware and software?</p>
</blockquote>

<p>Minor nit: If the algorithms got better then that is the software getting better so there is no "or".</p>

<p><a href="https://en.wikipedia.org/wiki/Moore%27s_law" rel="nofollow noreferrer">Moore's Law</a> tells us that processor speed will double roughly every 18 months. That means it has doubled roughly 13 times in 20 years. That makes modern processors somewhere in the region of 8,000 times faster. So, far and away the biggest improvement in engine performance is due to faster hardware.</p>

<blockquote>
  <p>If the former, are these algorithmic improvements public?</p>
  
  <p>And if so, what were the improvements? Where can I read about them?</p>
</blockquote>

<p>Well, it wasn't the former, it was the latter. Nevertheless the improvements are mostly open source and freely visible by <a href="https://stockfish.s3.amazonaws.com/stockfish-8-src.zip" rel="nofollow noreferrer">downloading the sources for engines like Stockfish</a>. Perhaps also worth giving the <a href="https://stockfishchess.org/download/" rel="nofollow noreferrer">general Stockfish download link</a> since the specific source code link will likely expire when version 9 comes out.</p>
 12449: <p>I can't speak for the algorithm used for Deep Blue, but I'm going to try and explain the improvements in chess programming.  Speed is the greatest improvement.  Deep Blue used multi-processor dedicated computers, so a comparison isn't really possible.</p>

<p><a href="https://chessprogramming.wikispaces.com/" rel="nofollow noreferrer">https://chessprogramming.wikispaces.com/</a> is a great source, but it's hard to navigate.  </p>

<p>There are 3 main functions that are tweaked to improve a chess engine are the evaluation, move generation, and search functions.</p>

<p>Evaluation is the hardest to program, as there are many exceptions to the rules.  With hard drive space getting cheaper, the eval function allows for more exceptions to be evaluated.</p>

<p>Move generation, along with making and unmaking a move, consumes a lot of memory because it has to be preformed so many times.  The most common generation functions are mailbox, bitboard, 0x88, 8x8, extended boards (10x10, 10x12), and a predetermined move array/table  (*I use an indexed move table).  Current opinion is that bitboards are the faster, and using magic bitboards speed this by up to 30%.  Dr. Robert Hyatt, professor and creator of cratfy chess engine, claims no significant speed increase.</p>

<p>The early search function was the primitive min-max functions.  Basically were you try to maximize the score of the side to move and minimize the opponent's score.  Alpha-Beta was the first improvement.  They reduced the number of moves being searched by transposition table, cut-off values, aspiration windows, and history heuristics.  These are depth-first searches.  There is also the internal iterative deepening search which tries to search the "best" move(s) the deepest hoping that searching other moves will prove to be fruitless.</p>

<p>NOTE:  My index table.
GNUChess and Jester both use an index array to generate their moves.  They initialize the engine by filling array with possible moves.  The take the six pieces and compute the legal moves that are available from each square.  So each piece had a [64][8] array.  I took this idea and compressed it to two indexes and a table.  The table holds a value which tells if the 16 moves are possible, one index holds the offset of the move, and the other holds the mask.</p>

<p>offset[] = {-8, -1, 1, 8, -9, -7, 7, 9, -17, -15, -10, -6, 6, 10, 15, 17};</p>

<p>mask[]   = {1, 2, 4, 8, 16, 32, 64, 128, 256, ...};</p>

<p>Then the generation of a sliding move is as easy as looking up the validity of it's mask in it's allowable offsets against the move table.</p>
 4588: <p>Maybe you can take a look at <a href="http://www.talkchess.com/forum/index.php" rel="noreferrer">TalkChess</a>, a forum dedicated to computer chess. I found a recent thread that might be interesting for you: <a href="http://talkchess.com/forum/viewtopic.php?t=62868" rel="noreferrer">Progress in 30 years by four intervals of 7-8 years</a></p>

<p>A couple of matches between (former) top engines are played on the <em>same hardware</em>. The test suggests that in the recent years (2002-2017), the gain is mainly made by software improvements. In the test, Stockfish (2017) scored an impressive 94/100 against RobboLito (2009), while RobboLito, on its turn, crushed Shredder (2002) with 92/100.</p>

<p>An important remark: as parallel computing is not implemented in the older engines, the test was performed on a single core. As a result, the hardware gain by parallel machines is not measured. On the other hand, you could argue that parallel computing is also a software gain: it is not easy to design and implement an efficient and well-scaling parallelization for the search algorithm.</p>

<p>The <a href="https://stockfishchess.org/" rel="noreferrer">Stockfish</a> engine is open source, so the algorithmic improvements are public. A lot of documentation can be found on <a href="https://chessprogramming.wikispaces.com" rel="noreferrer">https://chessprogramming.wikispaces.com</a></p>
