Finding Snakes in a Matrix
Python 2, 275 271 264 249 bytes


Saved four bytes by replacing -1 with H and removing one slicing operation ([:]).
Saved seven bytes thanks to Halvard Hummel; removing yet another slicing operation ([:]), using multiple-target assignment to give a visited entry a value v not in "01" (S=S[1:];M[y][x]=H; -> S=M[y][x]=S[1:];) and switching from a ternary if/else to a simple logical or (any(...)if S else 1 -> not S or any(...)).
If you somewhat extend your definition of truthy and falsey, you could allow this 257 byte long solution. It raises an exception (ZeroDivisionError) when the snake is found and returns an empty list ([]) when there is no snake to be found, which are two distinct behaviours.
Saved fourteen bytes thanks to user202729; golfing two array deep copies
Saved a byte; golfed not S or to S&lt;[1]or ~ S==[]or.




lambda M,S,w,h:any(H(eval(`M`),S,w,h,x,y)for y in range(h)for x in range(w)if S[0]==M[y][x])
def H(M,S,w,h,x,y):S=M[y][x]=S[1:];return S&lt;[1]or any(H(eval(`M`),S,w,h,x+X,y+Y)for X,Y in[(~0,0),(1,0),(0,~0),(0,1)]if~0&lt;x+X&lt;w&gt;0&lt;=y+Y&lt;h!=S[0]==M[y+Y][x+X])


Try it online!

Explanation

Lambda function which takes in the matrix as a two-dimensional list of strings (either "0" or "1"), the snake as a one-dimensional list and the matrix dimensions as two integers.
The lambda function searches the matrix for entries that match the snake's first element. For every found match, it calls H with a deep copy of the matrix, no copy of the snake, the matrix dimensions and the match's position.  

When H is called, it removes S' first entry and sets the given position's matrix entry to something else than "0", "1". If S' length is zero, it returns True; as it calls itself recursively, the snake was found somewhere in the matrix.
If S' length is non-zero, it loops through the four cardinal directions, tests if that position is in the matrix, compares the matrix' element at that position with the first element of S and -- if it matches -- calls itself recursively.
H's return values are funneled up the stack frames, always checking if at least one function found a possible snake.

Formatted Output

I have augmented my program to also output the path the snake slithers (if there is one). It uses the same ASCII output design as the question. TIO link.